Spectral Analysis of Stochastic Neural Oscillators

随机神经振荡器的谱分析

• 批准号: 1413770
• 负责人: Peter Thomas
• 金额: $ 23万
• 依托单位: Case Western Reserve University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2019-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1413770&HistoricalAwards=false
• 关键词: Spectral; Analysis; Stochastic; Neural; Oscillators

Many aspects of brain function involve the coordinated activity of millions of nerve cells. Individually, these cells can behave like tiny clocks, emitting a steady stream of pulses that communicate with each other. The synchronization of these oscillators ("clocks") plays a role in both healthy and diseased brain states. Deepening the understanding of how groups of brain cells synchronize ("tick" together) or desynchronize ("tick" separately) will deepen the understanding of the brain systems underlying motor control, epilepsy, breathing, information processing, and cognition. This project addresses a conceptual gap in the theoretical understanding of oscillating nerve cells. Today, most theories about how nerve cells synchronize rely on the assumption that each cell behaves with nearly impeccable precision. However, real nerve cells have stochastic variability, and their behavior is partly irregular and unpredictable. In constructing mathematical models of nerve cells that can account for variability, mathematically challenging problems arise. Solving these mathematical problems can improve the ability to quantitatively describe the clocklike behavior of individual nerve cells. This project will contribute to the BRAIN Initiative by providing deeper insight into many nervous system disorders.This project addresses a mathematical problem at the foundations of computational neuroscience and neurophysiology: the reduction of oscillatory systems to a phase oscillator description, in the presence of noise. The asymptotic phase of a deterministic dynamical system possessing a stable limit cycle was introduced in mathematical biology in the 1970s. Today, the biological significance of phase oscillator models is difficult to overstate. The fundamental notions of phase and phase resetting depend on classical results of invariant manifold theory for systems of deterministic differential equations. But noise is ubiquitous in biological dynamics. In recasting models of biological oscillators to take into account random fluctuations, the classical "asymptotic phase" is no longer well defined. This project develops a new definition for the asymptotic phase of a noisy oscillator, defined in terms of the complex eigenfunctions of the forward Kolmogorov operator, describing the evolution of the density on the state space. This "stochastic asymptotic phase" coincides with the classical phase in the case of vanishing noise intensities. However this new definition does not rely on the classical phase of a related deterministic system. Unlike the classical phase, it is equally well defined for systems that require noise for sustained oscillation.

大脑功能的许多方面涉及数百万神经细胞的协调活动。 单独来看，这些细胞的行为就像微型时钟一样，发出稳定的脉冲流，相互交流。 这些振荡器（“时钟”）的同步在健康和患病的大脑状态中都起着作用。 加深对脑细胞群如何同步（一起“滴答”）或去同步（分开“滴答”）的理解，将加深对运动控制、癫痫、呼吸、信息处理和认知背后的大脑系统的理解。 该项目解决了振荡神经细胞理论理解中的概念空白。 今天，大多数关于神经细胞如何同步的理论都依赖于这样一个假设，即每个细胞的行为几乎都是无可挑剔的。 但是，真实的神经细胞具有随机变异性，它们的行为部分是不规则的和不可预测的。 在构建可以解释变异性的神经细胞数学模型时，出现了数学上具有挑战性的问题。 解决这些数学问题可以提高定量描述单个神经细胞时钟行为的能力。 该项目将通过提供对许多神经系统疾病的更深入的了解来为BRAIN Initiative做出贡献。该项目解决了计算神经科学和神经生理学基础上的数学问题：在存在噪声的情况下，将振荡系统简化为相位振荡器描述。具有稳定极限环的确定性动力系统的渐近相是20世纪70年代在数学生物学中引入的。今天，相位振荡器模型的生物学意义很难夸大。 相位和相位重置的基本概念依赖于确定性微分方程系统的不变流形理论的经典结果。但噪声在生物动力学中无处不在。 在考虑随机波动的生物振荡器模型的重铸中，经典的“渐近相位”不再被很好地定义。这个项目开发了一个新的定义的噪声振荡器的渐近相位，定义在前向Kolmogorov算子的复杂的本征函数，描述状态空间上的密度的演变。这个“随机渐近阶段”与经典阶段的情况下，消失的噪声强度。然而，这个新的定义并不依赖于一个相关的确定性系统的经典阶段。与经典相位不同，它同样适用于需要噪声来维持振荡的系统。